Question

A zoo is designing a giant bird cage consisting of a cylinder of radius rr feet and height hh feet with a hemisphere on top (no bottom). The material for the hemisphere costs 2020 per square foot and the material for the cylindrical sides costs 1010 per square foot; the zoo has a budget of 45004500. Find the values of rr and hh giving the birds the greatest space inside assuming the zoo stays within its budget. Note: surface of a cylinder's side 2πrh2πrh, surface of a sphere 4πr24πr2, volume of a cylinder's side πr2hπr2h, volume of a sphere 43πr343πr3

Answer 1

To maximize the space inside the bird cage while staying within the budget, we need to optimize the volume function and consider the cost constraints. By taking the derivative of the volume function with respect to r and solving for r, we can find the radius that maximizes the volume. Substituting this radius into the equation for h, we can determine the height that corresponds to the maximum volume.

Step-by-step explanation:

To find the values of rr and hh giving the birds the greatest space inside the budget, we need to determine the constraints and optimize the volume function. The volume of the cylindrical part is given by Vc = πr^2h, and the volume of the hemisphere is given by Vh = (2/3)πr^3. The cost of the cylindrical part is 1010 per square foot, and the cost of the hemisphere is 2020 per square foot. The total cost must not exceed the budget of 4500.

Using the cost constraint, we can express the total cost as a function of r and h: C = 1010(2πrh) + 2020(2πr^2). We can rearrange this equation to express h in terms of r: h = (4500 - 4040πr^2) / (2020πr).

Substituting the expression for h in the volume equation, we get V = πr^2((4500 - 4040πr^2) / (2020πr)). Simplifying this expression, we obtain V = (4500 - 4040πr^2) / 2020.

To find the maximum volume, we can take the derivative of V with respect to r and set it equal to zero. Solving for r, we get r = 1.414. Substituting this value back into the expression for h, we find h = 1.414. Therefore, the values of rr and hh that give the birds the greatest space inside the budget are r = 1.414 feet and h = 1.414 feet.

Answer 2

r = 42,32 ft

h = 84.8 ft

Step-by-step explanation:

We are going to apply Lagrange multipliers method

The greatest space means maximum volume

V(cage) = Vol. of the cylinder + volume of the hemisphere

V(cylinder ) = π*r²*h

V(sphere) = (4/3)*π*r² ⇒ V(hemisphere) = (2/3)*π*r³

V(cage) = π*r²*h + (2/3)*π*r³

Associated costs:

Costs = cost of cylinder + cost of hemisphere

Area of the cylinder = Lateral area ( no bottom no top)

Area of the cylinder = 2*π*r*h

Area of hemisphere = 2*π*r²

A(r,h) = 2*π*r*h + 2*π*r²

C(r , h ) = 10* 2*π*r*h + 20* 2*π*r² C(r , h ) = 20*π*r*h + 40*π*r²

4500 = 20*π*r*h + 40*π*r²

20*π*r*h + 40*π*r² - 4500 = 0 20*π*r*h + 40*π*r² - 4500 = g(r,h)

V(cage) = π*r²*h + (2/3)*π*r³

δV/δr = 2*r*π*h + 2*π*r² δg(r,h)/δr = 20*π*h + 80*π*r

δV/δh = π*r² δg(r,h)/δh = 20*π*r

δV/δr = λ* δg(r,h)/δr

2*r*π*h + 2*π*r² = λ* 20*π*h + 80*π*r

2*r*π* ( h + r ) = 20*π* λ* ( h + 4*r)

r* ( h + r ) = 10*λ* ( h + 4*r) (1)

δV/δh = λ* δg(r,h)/δh

π*r² = 20*λ*π*r r = 20*λ (2)

20*π*r*h + 40*π*r² - 4500 = 0 (3)

We need to sole the system of equation 1 ; 2 ; 3

r = 20*λ plugging that value in equation 1

20*λ ( h + 20*λ ) = 10*λ* ( h + 4*r)

2( h + 20*λ ) = ( h + 4*20*λ)

2*h + 40*λ = h + 80*λ

h = 40*λ

20*π*r*h + 40*π*r² - 4500 = 0

20*π*20*λ*40*λ + 40*π+400λ² - 4500 = 0

16000*π*λ² + 16000*π*λ² = 4500

32000*π*λ² = 4500

320*π*λ² = 4500

λ² = 4500/1004,8 λ² = 4.48 λ = 2.12

Then

r = 20* λ r = 42,32 ft

h = 40* λ h = 84.8 ft